The parameters that measure the resistance to compression and resistance to shear of a solid are called elastic moduli and, together with density, determine two observables: the speed of propagation of stress waves (some of which are sound waves, some of which are shear waves) and the mechanical resonances. Resonances are most easily understood by thinking of the ringing of a bell. The tones produced by the bell are determined by the shape of the bell and the material from which it is made. Generally, if the shape is known, the tones of the bell can be used to determine the elastic moduli, or stiffnesses, of the material, and vice versa. Through the elastic stiffnesses, the wave speeds and resonances provide a measure of two fundamental thermodynamic quantities: internal energy and free energy. Unlike most of the properties used to characterize condensed matter, the elastic moduli are tensors (there can be up to 21 different moduli) containing a wealth of detail that provide one of the most revealing probes of solids. Elastic moduli also determine the frequencies of low-frequency phonons. Phonons are the quanta of vibration in solids. They can be thermally excited, and contribute to the internal energy and heat capacity. A phonon is a resonance; the lowest-energy phonons have wavelengths that approach the size of a specimen and are measured directly by resonance ultrasound methods. Because of the connections to fundamental thermodynamic properties, ultrasound measurements in condensed matter are so important that hundreds of useful approaches have been developed.

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